\newproblem{lay:4_5_5}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.5.5}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Ana Pe\~na Gil, Jan. 19th 2014} \\}{}

  % Problem statement
	Find a basis for the subspace below and state its dimension
	\begin{center}
		\[S=\left\{\begin{pmatrix}p-2q\\2p+5r\\-2q+2r\\-3p+6r\end{pmatrix}\quad \forall p,q,r\in\mathbb{R}\right\}\]
	\end{center}
}{
  % Solution
	We may write the set as
	\begin{center}
		\[S=\left\{p\begin{pmatrix}1\\2\\0\\-3\end{pmatrix}+q\begin{pmatrix}-2\\0\\-2\\0\end{pmatrix}+r\begin{pmatrix}0\\5\\2\\6\end{pmatrix}\quad \forall p,q,r\in\mathbb{R}\right\}\]\\
	\end{center}
	Thus, a basis is given by the vectors
	\begin{center}
		\[\mathrm{Basis}\{S\}=\left\{\begin{pmatrix}1\\2\\0\\-3\end{pmatrix},\begin{pmatrix}-2\\0\\-2\\0\end{pmatrix},\begin{pmatrix}0\\5\\2\\6\end{pmatrix}\right\}\] \\
	\end{center}
	Since the basis has three vectors, the dimension of $S$ is 3.\\
}
\useproblem{lay:4_5_5}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
